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Physics 2

Electric charge and Coulomb's law

Properties of charge, conductors and insulators, electrostatic induction, Coulomb's law and superposition with an example, comparison with gravity and electrostatic discharge.

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Physics 2 is electromagnetism: the part of physics on which all of electronics rests. It starts from the simplest quantity, electric charge, and from the force between two charges, Coulomb’s law. In this lesson we look at the properties of charge, Coulomb’s law with the superposition principle, and the comparison with gravity.

Electric charge

Rub a plastic rod with a woollen cloth and the rod attracts bits of paper. There are two kinds of charge, conventionally called positive and negative: like charges repel, unlike charges attract. The basic properties:

  • charge is conserved: in an isolated system the total charge does not change. Rubbing does not create charge, it moves electrons from one body to another;
  • charge is quantised: every charge is an integer multiple of the elementary charge e = 1.602\times 10^{-19} C. The electron has charge -e, the proton +e;
  • the unit is the coulomb (C), the charge carried in one second by a current of one ampere. In electrostatics a coulomb is huge: about 6.24\times 10^{18} electrons.

Conductors and insulators

In conductors (metals, ionic solutions) some charges are free to move: in metals they are the conduction electrons, about one per atom in copper. In insulators (glass, plastic, dry air) charges stay bound to atoms. Semiconductors sit in between, and the number of free charges is controlled by doping and temperature: the basis of diodes and transistors.

Electrostatic induction. Bringing a charged rod near a neutral conductor, the free electrons move: the near side takes the opposite charge and the far side the same charge. The conductor stays neutral but is attracted, because the opposite charge is closer. It is also why a rubbed balloon sticks to a wall.

Coulomb’s law

Two point charges q_1 and q_2 a distance r apart exert on each other a force along the line joining them, of magnitude

    \[ F = k\,\frac{|q_1\,q_2|}{r^2},\qquad k = \frac{1}{4\pi\varepsilon_0} \approx 8.99\times 10^9\ \text{N·m}^2/\text{C}^2 \]

where \varepsilon_0 = 8.854\times 10^{-12} F/m is the permittivity of free space.

Forces between two like charges, which repel, and two unlike charges, which attract
Top: two positive charges repel; bottom: a positive and a negative charge attract. In both cases the two forces are equal and opposite (third law) and directed along the line joining the charges.

In vector form, the force q_1 exerts on q_2 is

    \[ \vec F_{12} = \frac{1}{4\pi\varepsilon_0}\,\frac{q_1 q_2}{r^2}\,\hat r_{12} \]

with \hat r_{12} the unit vector from q_1 to q_2. If q_1 q_2 \gt 0 the force is repulsive, if negative attractive. The 1/r^2 dependence is the same as gravitation, and like gravitation it has been tested with extreme precision: the exponent is 2 to within one part in 10^{16}.

The superposition principle

With several charges, the force on one of them is the vector sum of the forces exerted by the others, each computed as if the others were not there.

Example. Three charges along the x axis: q_1 = +2 µC at x = 0, q_2 = -3 µC at x = 1 m, q_3 = +1 µC at x = 2 m. Force on q_3:

  • from q_1 (repulsive, to the right): 8.99\times 10^9\cdot\frac{2\times 10^{-6}\cdot 10^{-6}}{4} = 4.5 mN;
  • from q_2 (attractive, to the left): 8.99\times 10^9\cdot\frac{3\times 10^{-6}\cdot 10^{-6}}{1} = 27 mN.

The resultant is about 22.5 mN to the left.

Electricity versus gravity

In the hydrogen atom, electron and proton are about 5.3\times 10^{-11} m apart. The electric force is about 8.2\times 10^{-8} N, the gravitational force 3.6\times 10^{-47} N: the ratio is about 10^{39}. Gravity rules the sky only because large bodies are almost perfectly neutral: positive and negative charges cancel. A tiny imbalance would produce enormous forces: two people a metre apart, each with 1% more electrons than protons, would repel each other with a force comparable to the weight of the Earth.

Continuous charge distributions

For extended charged bodies, split the charge into elements dq and integrate. Depending on the geometry one uses a linear density \lambda (C/m, for a wire), a surface density \sigma (C/m², for a sheet) or a volume density \rho (C/m³): dq = \lambda\,dl, \sigma\,dA or \rho\,dV. The integrals are those of Analysis 2, and the choice of coordinates follows the symmetry of the body.

A practical note: electrostatic discharge

Walking on carpet can charge your body to a few kilovolts. Touching an integrated circuit, the discharge (ESD) crosses junctions and oxides only a few nanometres thick and damages them, sometimes with no visible effect at first. That is why labs use grounded wrist straps, dissipative mats and antistatic bags, and why chips have protection diodes on every pin.

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